Prime and Almost Prime Points on Homogeneous Varieties

Let V be the common zero set of a family of r integral homogeneous forms of degree k in n variables.

We show, assuming V is smooth and n is sufficiently large with respect to r and k, that V contains the expected number of prime points. A crucial ingredient proof is a regularization process of the system of polynomials, motivated by similar arguments in combinatorics. This however makes the the bounds on n tower-exponential type in the parameters r and k.

Next, we study the number of almost prime points on V, where one can get lower bounds under less restrictive conditions on the number of variables n, which are in agreement with the case of integer solutions. The argument combines classical Hardy-Littlewood type methods with techniques developed by Goldston-Yildirim-Pintz for short divisor sums.